Section 2.
5 Continuity
Definition of Continuity: A function 𝑓 is continuous at a number 𝑎 if _____________________________________
This Definition implies three things:
1) ____________________________________________________________________________________
2) ____________________________________________________________________________________
3) ____________________________________________________________________________________
Discontinuities: There are three types of discontinuities
1) ___________________________________
2) ___________________________________
3) ___________________________________
1
Theorem: If ___________________________________________________________________and 𝑐 is a constant,
then the following functions are also continuous at 𝑎:
1) ________________ 2) ________________
3) ________________ 4) ________________
Theorem: The following type of functions are continuous at every number in their domains:
1) Polynomials 2) Rational Functions 3) Root Functions
4) Trig Functions 5) Inverse Trig Functions 6) Exponential Functions
7) Logarithmic Functions
Example: From the graph of 𝑔, state the numbers at which 𝑔 is discontinuous. For each of these numbers.
determine whether 𝑔 is continuous from the right or from the left.
Example: Use the definition of continuity and the properties of limits to show that the function is continuous at the
given number.
! ! "#!
𝑔(𝑡) = $!"%
𝑎=2
2
Example: Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere.
& ! '(
&'$
𝑖𝑓 𝑥 < 2
1) 𝑓(𝑥) = ,𝑎𝑥 $ − 𝑏𝑥 + 3 𝑖𝑓 2 ≤ 𝑥 < 3
2𝑥 − 𝑎 + 𝑏 𝑖𝑓𝑥 ≥ 3
4 𝑖𝑓 𝑥 < −1
(S) 2) 𝑓(𝑥) = 5𝑎𝑥 − 𝑏 𝑖𝑓 − 1 ≤ 𝑥 < 1
6 𝑖𝑓𝑥 ≥ 1
Example: Suppose 𝑓 and 𝑔 are continuous functions such that 𝑔(2) = 6 and lim[3𝑓(𝑥) + 𝑓(𝑥)𝑔(𝑥)] = 36.
&→$
Find 𝑓(2).
3
(S) Example: The gravitational force exerted by the planet Earth on a unit mass at a distance 𝑟 from the center of
the planet is
𝐺𝑀𝑟
𝑖𝑓 𝑟 < 𝑅
𝐹(𝑟) = , 𝑅
+
𝐺𝑀
𝑖𝑓 𝑟 ≥ 𝑅
𝑟$
where 𝑀 is the mass of the Earth, 𝑅 is the radius, and 𝐺 is the gravitational constant. Is 𝐹 a continuous function of
𝑟?
Theorem: If 𝑓 is continuous at ____________ and __________________________________ then
____________________________________, or _____________________________________________________.
Theorem: If 𝑔 is continuous at 𝑎 and 𝑓 is continuous at 𝑔(𝑎) then the composite function 𝑓 ∘ 𝑔 given by
(𝑓 ∘ 𝑔) = 𝑓(𝑔(𝑥)) is continuous at 𝑎.
The Intermediate Value Theorem: Suppose that 𝑓 is a continuous function on the closed interval from
________________ and let ______________ be any number between ______________ and _____________
where ________________________. Then __________________________________________________________
________________________________________________
________________________________________________
